PLANE WAVES 9 



A solution of this equation for a simple harmonic wave traveling in the 

 positive X direction is 



^ = A cos k{ct — x) 1.17 



where A = amplitude of 0, 



k = 27r/X 

 X = wavelength. 



A. Particle Velocity in a Plane Wave. — ■ The particle velocity, «, em- 

 ploying equations 1.11 and 1.17 is 



u = ■ — = kA sin k{ct — x) 1.18 



dx 



The particle velocity in a sound wave is the instantaneous velocity of a 

 given infinitesimal part of the medium, with reference to the medium as a 

 whole, due to the passage of the sound wave. 



B. Pressure in a Plane Wave. — From equations 1.9, 1.13 and 1.15 the 

 following relation may be obtained 



^-^^-cH 1.19 



a/ 



The condensation in a plane wave from equations 1.19 and 1.17 is 



given by 



Ak 

 s = — sin kc {ct — x) 1 .20 



c 



From equations 1.9 and 1.15 the following relation may be obtained 



p = c'-ps 1.21 



Then, from equations 1.20 and 1.21 the pressure in a plane wave is 



p = kcpA s\n k{ct — x) 1.22 



Note: the particle velocity, equation 1.18, and the pressure, equation 

 1.22 are in phase in a plane wave. 



C. Particle Amplitude in a Plane Wave. — The particle amplitude of a 

 sound wave is the maximum distance that the vibrating particles of the 

 medium are displaced from the position of equilibrium. 



From equation 1.18 the particle velocity is 



i^ u = kA sin k{ct - x) 1.23 



where ^ = amphtude of the particle from its equilibrium position, in 

 centimeters. 



